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TRICHOTOMY, STABILITY, AND OSCILLATION OF A FUZZY DIFFERENCE EQUATION G. STEFANIDOU AND G. PAPASCHINOPOULOS Received 10 November 2003 We study the trichotomy character, the stability, and the oscillatory behavior of the positive solutions of a fuzzy diﬀerence equation. 1. Introduction Diﬀerence equations have already been successfully applied in a number of sciences (for a detailed study of the theory of diﬀerence equations and their applications, see [1, 2, 7, 8, 11]. The problem of identifying, modeling, and solving a nonlinear diﬀerence equation concerning a real-world phenomenon from experimental input-output data, which is uncertain, incomplete, imprecise, or vague, has been attracting increasing attention in recent years. In addition, nowadays, there is an increasing recognition that for understanding vagueness, a fuzzy approach is required. The eﬀect is the introdution and the study of the fuzzy diﬀerence equations (see [3, 4, 13, 14, 15]). In this paper, we study the trichotomy character, the stability, and the oscillatory behavior of the positive solutions of the fuzzy diﬀerence equation k ci xn− pi , d j =1 j xn−q j xn+1 = A + mi=1 (1.1) where k,m ∈ {1,2,... }, A,ci ,d j , i ∈ {1,2,...,k}, j ∈ {1,2,...,m}, are positive fuzzy numbers, pi , i ∈ {1,2,...,k}, q j , j ∈ {1,2,...,m}, are positive integers such that p1 < p2 < · · · < pk , q1 < q2 < · · · < qm , and the initial values xi , i ∈ {−π, −π + 1,...,0}, where π = max pk , qm , (1.2) are positive fuzzy numbers. Studying a fuzzy diﬀerence equation results concerning the behavior of a related family of systems of parametric ordinary diﬀerence equations is required. Some necessary results Copyright © 2004 Hindawi Publishing Corporation Advances in Diﬀerence Equations 2004:4 (2004) 337–357 2000 Mathematics Subject Classification: 39A10 URL: http://dx.doi.org/10.1155/S1687183904311015 338 Fuzzy diﬀerence equations concerning the corresponding family of systems of ordinary diﬀerence equations of (1.1) have been proved in [16] and others are given in this paper. 2. Preliminaries We need the following definitions. For a set B, we denote by B the closure of B. We say that a fuzzy set A, from R+ = (0, ∞) into the interval [0,1], is a fuzzy number, if A is normal, convex, upper semicontinu ous (see [14]), and the support suppA = a∈(0,1] [A]a = {x : A(x) > 0} is compact. Then from [12, Theorems 3.1.5 and 3.1.8], the a-cuts of the fuzzy number A, [A]a = {x ∈ R+ : A(x) ≥ a}, are closed intervals. We say that a fuzzy number A is positive if suppA ⊂ (0, ∞). It is obvious that if A is a positive real number, then A is a positive fuzzy number and [A]a = [A,A], a ∈ (0,1]. In this case, we say that A is a trivial fuzzy number. We say that xn is a positive solution of (1.1) if xn is a sequence of positive fuzzy numbers which satisfies (1.1). A positive fuzzy number x is a positive equilibrium for (1.1) if k ci x . d j =1 j x x = A + mi=1 (2.1) Let E, H be fuzzy numbers with [E]a = [El,a ,Er,a ], [H]a = [Hl,a ,Hr,a ], a ∈ (0,1]. (2.2) According to [10] and [13, Lemma 2.3], we have that MIN{E,H } = E if El,a ≤ Hl,a , Er,a ≤ Hr,a , a ∈ (0,1]. (2.3) Moreover, let ci , fi ,d j ,g j , i = 1,2,...,k, j = 1,2,...,m, be positive fuzzy numbers such that for a ∈ (0,1], = ci,l,a ,ci,r,a , d j a = d j,l,a ,d j,r,a , k ci E = mi=1 , ci a j =1 d j = fi,l,a , fi,r,a , g j a = g j,l,a ,g j,r,a , k fi H = mi=1 . fi a j =1 g j (2.4) (2.5) We will say that E is less than H and we will write E≺H (2.6) if k i=1 supa∈(0,1] ci,r,a m j =1 inf a∈(0,1] d j,l,a k inf a∈(0,1] fi,l,a < mi=1 . j =1 supa∈(0,1] g j,r,a (2.7) G. Stefanidou and G. Papaschinopoulos 339 In addition, we will say that E is equal to H and we will write . E=H if E ≺ H, H ≺ E, (2.8) which means that for i = 1,2,...,k, j = 1,2,...,m, and a ∈ (0,1], ci,l,a = ci,r,a , fi,l,a = fi,r,a , d j,l,a = d j,r,a , g j,l,a = g j,r,a , (2.9) and so El,a = Er,a = Hl,a = Hr,a , a ∈ (0,1], (2.10) which imples that E, H are equal real numbers. For the fuzzy numbers E, H, we give the metric (see [9, 17, 18]) D(E,H) = supmax El,a − Hl,a , Er,a − Hr,a , (2.11) where sup is taken for all a ∈ (0,1]. The fuzzy analog of boundedness and persistence (see [5, 6]) is given as follows: we say that a sequence of positive fuzzy numbers xn persists (resp., is bounded) if there exists a positive number M (resp., N) such that supp xn ⊂ [M, ∞) resp., suppxn ⊂ (0,N] , n = 1,2,.... (2.12) In addition, we say that xn is bounded and persists if there exist numbers M,N ∈ (0, ∞) such that suppxn ⊂ [M,N], n = 1,2,.... (2.13) Let xn be a sequence of positive fuzzy numbers such that xn a = Ln,a ,Rn,a , a ∈ (0,1], n = 0,1,..., (2.14) and let x be a positive fuzzy number such that [x]a = [La ,Ra ], a ∈ (0,1]. (2.15) We say that xn nearly converges to x with respect to D as n → ∞ if for every δ > 0, there exists a measurable set T, T ⊂ (0,1], of measure less than δ such that limDT xn ,x = 0, as n −→ ∞, (2.16) where DT xn ,x = sup a∈(0,1]−T max Ln,a − La , Rn,a − Ra . If T = ∅, we say that xn converges to x with respect to D as n → ∞. (2.17) 340 Fuzzy diﬀerence equations Let X be the set of positive fuzzy numbers. Let E,H ∈ X. From [18, Theorem 2.1], we have that El,a , Hl,a (resp., Er,a , Hr,a ) are increasing (resp., decreasing) functions on (0,1]. Therefore, using the definition of the fuzzy numbers, there exist the Lebesque integrals J El,a − Hl,a da, J Er,a − Hr,a da, (2.18) where J = (0,1]. We define the function D1 : X × X → R+ such that D1 (E,H) = max El,a − Hl,a da, J J Er,a − Hr,a da . (2.19) If D1 (E,H) = 0, we have that there exists a measurable set T of measure zero such that El,a = Hl,a , Er,a = Hr,a ∀a ∈ (0,1] − T. (2.20) We consider, however, two fuzzy numbers E, H to be equivalent if there exists a measurable set T of measure zero such that (2.20) hold and if we do not distinguish between equivalence of fuzzy numbers, then X becomes a metric space with metric D1 . We say that a sequence of positive fuzzy numbers xn converges to a positive fuzzy number x with respect to D1 as n → ∞ if limD1 xn ,x = 0, as n −→ ∞. (2.21) We define the fuzzy analog for periodicity (see [11]) as follows. A sequence {xn } of positive fuzzy numbers xn is said to be periodic of period p if D xn+p ,xn = 0, n = 0,1,.... (2.22) Suppose that (1.1) has a unique positive equilibrium x. We say that the positive equithere exists a δ = δ() such that for everyposlibrium x of (1.1) is stable if for every > 0, itive solution xn of (1.1) which satisfies D x−i ,x ≤ δ, i = 0,1,...,π, we have D xn ,x ≤ for all n ≥ 0. Moreover, we say that the positive equilibrium x of (1.1) is nearly asymptotically stable if it is stable and every positive solution of (1.1) nearly tends to the positive equilibrium of (1.1) with respect to D as n → ∞. Finally, we give the fuzzy analog of the concept of oscillation (see [11]). Let xn be a sequence of positive fuzzy numbers and let x be a positive fuzzy number. We say that xn oscillates about x if for every n0 ∈ N, there exist s,m ∈ N, s,m ≥ n0 , such that MIN xm ,x = xm , MIN xs ,x = x (2.23) or MIN xm ,x = x, MIN xs ,x = xs . (2.24) G. Stefanidou and G. Papaschinopoulos 341 3. Main results Arguing as in [13, 14, 15], we can easily prove the following proposition which concerns the existence and the uniqueness of the positive solutions of (1.1). Proposition 3.1. Consider (1.1), where k,m ∈ {1,2,... }, A,ci ,d j , i ∈ {1,2,...,k}, j ∈ {1, 2,...,m}, are positive fuzzy numbers, and pi , q j , i ∈ {1,2,...,k}, j ∈ {1,2,...,m}, are positive integers. Then for any positive fuzzy numbers x−π ,x−π+1 ,...,x0 , there exists a unique positive solution xn of (1.1) with initial values x−π ,x−π+1 ,...,x0 . Now, we present conditions so that (1.1) has unbounded solutions. Proposition 3.2. Consider (1.1), where k,m ∈ {1,2,... }, A,ci ,d j , i ∈ {1,2,...,k}, j ∈ {1, 2,...,m}, are positive fuzzy numbers, and pi , i ∈ {1,2,...,k}, q j , j ∈ {1,2,...,m}, are positive integers. If k ci G = mi=1 , A ≺ G, (3.1) j =1 d j then (1.1) has unbounded solutions. Proof. Let [A]a = Al,a ,Ar,a , a ∈ (0,1]. (3.2) From (2.4) and (3.2) and since A, ci , d j , i = 1,2,...,k, j = 1,2,...,m, are positive fuzzy numbers, there exist positive real numbers B, C, ai , ei , h j , b j , i = 1,2,...,k, j = 1,2,...,m, such that B = inf Al,a , C = sup Ar,a , ei = sup ci,r,a , h j = inf d j,l,a , a∈(0,1] a∈(0,1] a∈(0,1] a∈(0,1] ai = inf ci,l,a , a∈(0,1] b j = sup d j,r,a . (3.3) a∈(0,1] Let xn be a positive solution of (1.1) such that (2.14) hold and the initial values xi , i = −π, −π + 1,...,0, are positive fuzzy numbers which satisfy xi a = Li,a ,Ri,a , i = −π, −π + 1,...,0, a ∈ (0,1] (3.4) and for a fixed ā ∈ (0,1], the relations Ri,ā > Z2 , W −C Li,ā < W, i = −π, −π + 1,...,0, (3.5) are satisfied, where k ei Z = mi=1 , j =1 h j k ai W = mi=1 . j =1 b j (3.6) 342 Fuzzy diﬀerence equations Using [15, Lemma 1], we can easily prove that Ln,a , Rn,a satisfy the family of systems of parametric ordinary diﬀerence equations k ci,l,a Ln− pi ,a , Ln+1,a = Al,a + mi=1 d j =1 j,r,a Rn−q j ,a k Rn+1,a = Ar,a + ci,r,a Rn− pi ,a mi=1 , j =1 d j,l,a Ln−q j ,a n = 0,1,.... (3.7) Since (3.1) holds, it is obvious that k ci,r,ā . d j =1 j,l,ā Al,ā < mi=1 (3.8) Using (3.8) and applying [16, Proposition 1] to the system (3.7) for a = ā, we have that lim Ln,ā=Al,ā , n→∞ lim Rn,ā = ∞. (3.9) n→∞ Therefore, from (3.9), the solution xn of (1.1) which satisfies (3.4) and (3.5) is un bounded. Remark 3.3. From the proof of Proposition 3.2, it is obvious that (1.1) has unbounded solutions if there exists at least one a ∈ (0,1] such that (3.8) holds. In the following proposition, we study the boundedness and persistence of the positive solutions of (1.1). Proposition 3.4. Consider (1.1), where k,m ∈ {1,2,... }, A,ci ,d j , i ∈ {1,2,...,k}, j ∈ {1,2,...,m}, are positive fuzzy numbers, and pi , i ∈ {1,2,...,k }, q j , j ∈ {1,2,...,m}, are positive integers. If either . A=G (3.10) G≺A (3.11) or holds, then every positive solution of (1.1) is bounded and persists. Proof. Firstly, suppose that (3.10) is satisfied; then A, ci , d j , i = 1,2,...,k, j = 1,2,...,m, are positive real numbers. Hence, for i = 1,2,...,k, j = 1,2,...,m, we get A = Al,a = Ar,a , ci = ci,l,a = ci,r,a , d j = d j,l,a = d j,r,a , k ci A = mi=1 . j =1 d j a ∈ (0,1], (3.12) (3.13) G. Stefanidou and G. Papaschinopoulos 343 Let xn be a positive solution of (1.1) such that (2.14) hold and let xi , i = −π, −π + 1,...,0, be the positive initial values of xn such that (3.4) hold. Then there exist positive numbers Ti , Si , i = −π, −π + 1,...,0, such that Ti ≤ Li,a ,Ri,a ≤ Si , i = −π, −π + 1,...,0. (3.14) Let (yn ,zn ) be the positive solution of the system of ordinary diﬀerence equations k ci yn− pi , d j =1 j zn−q j yn+1 = A + mi=1 k ci zn− pi , d j =1 j y n − q j zn+1 = A + mi=1 (3.15) with initial values (yi ,zi ), i = −π, −π + 1,...,0, such that yi = Ti , zi = Si , i = −π, −π + 1,...,0. Then from (3.14) and (3.15), we can easily prove that y1 ≤ L1,a , R1,a ≤ z1 , a ∈ (0,1], (3.16) n = 1,2,..., a ∈ (0,1]. (3.17) and working inductively, we take yn ≤ Ln,a , Rn,a ≤ zn , Since from (3.13) and [16, Proposition 3], (yn , zn ) is bounded and persists, from (3.17), it is obvious that xn is also bounded and persists. Now, suppose that (3.11) holds; then B > Z, C > W. (3.18) We concider the system of ordinary diﬀerence equations k ai y n − p i , j =1 b j zn−q j yn+1 = B + mi=1 k ei zn− pi , j =1 h j y n − q j zn+1 = C + mi=1 (3.19) where B,C,ai ,ei ,b j ,h j , i = 1,2,...,k, j = 1,2,...,m, are defined in (3.3). Let (yn ,zn ) be a solution of (3.19) with initial values (yi ,zi ), i = −π, −π + 1,...,0, such that yi = Ti , zi = Si , i = −π, −π + 1,...,0, where Ti ,Si , i = −π, −π + 1,...,0, are defined in (3.14). Arguing as above, we can prove that (3.17) holds. Since from (3.18) and [16, Proposition 3], (yn , zn ) is bounded and persists, then from (3.17), it is obvious that, xn is also bounded and persists. This completes the proof of the proposition. In what follows, we need the following lemmas. Lemma 3.5. Let ri ,s j , i = 1,2,...,k, j = 1,2,...,m, be positive integers such that r1 ,r2 ,...,rk ,s1 ,s2 ,...,sm = 1, (3.20) where (r1 ,r2 ,...,rk ,s1 ,s2 ,...,sm ) is the greatest common divisor of the integers ri ,s j , i = 1,2, ...,k, j = 1,2,...,m. Then the following statements are true. 344 Fuzzy diﬀerence equations (I) There exists an even positive integer w1 such that for any nonnegative integer p, there exist nonnegative integers αip , β j p , i = 1,2,...,k, j = 1,2,...,m, such that k αip ri + i =1 m β j p s j = w1 + 2p, p = 0,1,..., (3.21) j =1 where mj=1 β j p is an even integer. (II) Suppose that all ri , i = 1,2,...,k, are not even and all s j , j = 1,2,...,m, are not odd integers. Then there exists an odd positive integer w2 such that for any nonnegative integer p, there exist nonnegative integers γip , δ j p , i = 1,2,...,k, j = 1,2,...,m, such that k γip ri + i =1 m δ j p s j = w2 + 2p, p = 0,1,..., (3.22) j =1 where mj=1 δ j p is an even integer. (III) Suppose that all ri , i = 1,2,...,k, are not even and all s j , j = 1,2,...,m, are not odd integers. Then there exists an even positive integer w3 such that for any nonnegative integer p, there exist nonnegative integers ip , ξ j p , i = 1,2,...,k, j = 1,2,...,m, such that k ip ri + i =1 m ξ j p s j = w3 + 2p, p = 0,1,..., (3.23) j =1 where mj=1 ξ j p is an odd integer. (IV) There exists an odd positive integer w4 such that for any nonnegative integer p, there exist nonnegative integers λip , µ j p , i = 1,2,...,k, j = 1,2,...,m, such that k λip ri + i =1 where m j =1 µ j p m µ j p s j = w4 + 2p, p = 0,1,..., (3.24) j =1 is an odd integer. Proof. (I) Since(3.20) holds, there exist integers ηi , ι j , i = 1,2,...,k, j = 1,2,...,m, such that k ηi ri + i =1 m ι j s j = 1. (3.25) j =1 If for any real number a, we denote by [a] the integral part of a, we set for i = 2,3,...,k, j = 1,2,...,m, α1p = 2pη1 + 2 k ri + 2 i =2 αip = 2pηi + 2gip r1 , m j =1 sj − 2 k i=2 gip ri − 2 m j =1 β j p = 2pι j + 2h j p r1 , h j ps j , (3.26) G. Stefanidou and G. Papaschinopoulos 345 where gip = − pηi r1 + 1, hjp = − pι j i = 2,3,...,k, j = 1,2,...,m. + 1, r1 (3.27) Therefore, from (3.25) and (3.26), we can easily prove that αip ,β j p , i = 1,2,...,k, j = 1,2,...,m, which are defined in (3.26), are positive integers satisfying (3.21) for w1 = 2r1 k ri + i =2 m sj (3.28) j =1 and mj=1 β j p is an even number. (II) Firstly, suppose that one of ri , i = 1,2,...,k, is an odd positive integer and without loss of generality, let r1 be an odd positive integer. Relation (3.22) follows immediately if we set for i = 2,...,k and for j = 1,2,...,m, γ1p = α1p + 1, γip = αip , δ j p = β j p, w2 = w1 + r1 . (3.29) Now, suppose that ri , i = 1,2,...,k, are even positive integers; then from (3.20), one of s j , j = 1,2,...,m, is an odd positive integer and from the hypothesis, one of s j , j = 1,2,...,m, is an even positive integer. Without loss of generality, let s1 be an odd positive integer and s2 be an even positive integer. Relation (3.22) follows immediately if we set for i = 1,2,...,k and for j = 3,...,m, γip = αip , δ1p = β1p + 1, δ2p = β2p + 1, δ j p = β j p, w2 = w1 + s1 + s2 . (3.30) (III) Firstly, suppose that one of s j , j = 1,2,...,m, is an even positive integer and without loss of generality, let s1 be an even positive integer. Relation (3.23) follows immediately if we set for i = 1,2,...,k and j = 2,...,m, ip = αip , ξ1p = β1p + 1, ξ j p = β j p, w3 = w1 + s1 . (3.31) Now, suppose that s j , j = 1,2,...,m, are odd positive integers; then from the hypothesis, at least one of ri , i = 1,2,...,k, is an odd positive integer, and without loss of generality, let r1 be an odd integer. Relation (3.23) follows immediately if we set for i = 2,...,k, j = 2,3,...,m, 1p = α1p + 1, ip = αip , δ1p = β1p + 1, δ j p = β j p, w3 = w1 + s1 + r1 . (3.32) (IV) Firstly, suppose that at least one of s j , j = 1,2,...,m, is an odd positive integer and without loss of generality, let s1 be an odd positive integer. Relation (3.24) follows immediately if we set for i = 1,2,...,k, j = 2,3,...,m, λip = αip , µ1p = β1p + 1, µ j p = β j p, w4 = w1 + s1 . (3.33) 346 Fuzzy diﬀerence equations Now, suppose that s j , j = 1,2,...,m, are even positive integers; then from (3.20), at least one of ri , i = 1,2,...,k, is an odd positive integer, and without loss of generality, let r1 be an odd positive integer. Relation (3.24) follows immediately if we set for i = 2,3,...,k, j = 2,3,...,m, λ1p = α1p + 1, λip = αip , µ1p = β1p + r1 , µ j p = β j p, w4 = w1 + r1 s1 + 1 . (3.34) This completes the proof of the lemma. Lemma 3.6. Consider system (3.19), where B,C are positive constants such that k ei B = mi=1 , j =1 h j k ai C = mi=1 . (3.35) j =1 b j Then the following statements are true. (I) Let r be a common divisor of the integers pi + 1, q j + 1, i = 1,2,...,k, j = 1,2,...,m, such that pi + 1 = rri , i = 1,2,...,k, q j + 1 = rs j , j = 1,2,...,m; (3.36) then system (3.19) has periodic solutions of prime period r. Moreover, if all ri , i = 1,2,...,k, (resp., s j , j = 1,2,...,m) are even (resp., odd) positive integers, then system (3.19) has periodic solutions of prime period 2r. (II) Let r be the greatest common divisor of the integers pi + 1, q j + 1, i = 1,2,...,k, j = 1,2,...,m, such that (3.36) hold; then if all ri , i = 1,2,...,k, (resp., s j , j = 1,2,...,m) are even (resp., odd) positive integers, every positive solution of (3.19) tends to a periodic solution of period 2r; otherwise, every positive solution of (3.19) tends to a periodic solution of period r. Proof. (I) From relations (3.35), (3.36), and [16, Proposition 2], system (3.19) has periodic solutions of prime period r. Now, we prove that system (3.19) has periodic solutions of prime period 2r, if all ri , i = 1,2,...,k, (resp., s j , j = 1,2,...,m) are even (resp., odd) positive integers. Suppose first that pk < qm . Let (yn ,zn ) be a positive solution of (3.19) with initial values satisfying y−rsm +2rλ+ζ = y−r+ζ , z−rsm +2rλ+ζ = z−r+ζ , y−rsm +2rν+r+ζ = y−2r+ζ , z−rsm +2rν+r+ζ = z−2r+ζ , s −1 s −3 λ = 0,1,..., m , ν = 0,1,..., m , ζ = 1,2,...,r, 2 2 (3.37) and, in addition, for ζ = 1,2,...,r, y−2r+ζ > B, y−r+ζ > B, z−r+ζ = C y−2r+ζ , y−2r+ζ − B z−2r+ζ = C y−r+ζ . y−r+ζ − B (3.38) G. Stefanidou and G. Papaschinopoulos 347 From (3.19), (3.35), (3.36), (3.37), and (3.38), we get for ζ = 1,2,...,r, y−2r+ζ = y−2r+ζ , z−r+ζ y−r+ζ = B+C = y−r+ζ , z−2r+ζ z−2r+ζ = z−2r+ζ , y−r+ζ z−r+ζ = C+B = z−r+ζ . y−2r+ζ yζ = B + C zζ = C + B yr+ζ zr+ζ (3.39) Let a v ∈ 2,3,... . Suppose that for all u = 1,2,...,v − 1 and ζ = 1,2,...,r, we have y2ur+ζ = y−2r+ζ , z2ur+ζ = z−2r+ζ , y2ur+r+ζ = y−r+ζ , z2ur+r+ζ = z−r+ζ . (3.40) Then from (3.19), (3.35)–(3.40), we get for ζ = 1,2,...,r, y2vr+ζ = B + C y−2r+ζ = y−2r+ζ . z−r+ζ (3.41) Similarly, we can prove that for ζ = 1,2,...,r, z2vr+ζ = z−2r+ζ , y2vr+r+ζ = y−r+ζ , z2vr+r+ζ = z−r+ζ . (3.42) Therefore, from (3.39)–(3.42), we have that system (3.19) has periodic solutions of period 2r. Now, suppose that qm < pk . Let (yn ,zn ) be a positive solution of (3.19) such that the initial values satisfy relations (3.38) and for ω = 0,1,...,rk /2 − 1, θ = 1,2,...,2r, y−rrk +2rω+θ = y−2r+θ , z−rrk +2rω+θ = z−2r+θ . (3.43) Then arguing as above, we can easily prove that (yn ,zn ) is a periodic solution of period 2r. This completes the proof of statement (I). (II) Now, we prove that every positive solution of system (3.19) tends to a periodic solution of period κr, where 2 if ri , i = 1,2,...,k, are even, s j , j = 1,2,...,m, are odd, κ= 1 otherwise. (3.44) Let (yn ,zn ) be an arbitrary positive solution of (3.19). We prove that there exist the lim yκnr+i = i , n→∞ i = 0,1,...,κr − 1. (3.45) We fix a τ ∈ {0,1,...,κr − 1}. Since from [16, Proposition 3], the solution (yn ,zn ) is bounded and persists, we have liminf yκnr+τ = lτ ≥ B, n→∞ limsup yκnr+τ = Lτ < ∞, n→∞ liminf zκnr+τ = mτ ≥ C, n→∞ limsup zκnr+τ = Mτ < ∞. (3.46) n→∞ Therefore, from relations (3.19), (3.35), and (3.46), we take mτ = CLτ , Lτ − B lτ = BMτ . Mτ − C (3.47) 348 Fuzzy diﬀerence equations We prove that (3.45) is true for i = τ. Suppose on the contrary that lτ < Lτ . Then from (3.46), there exists an > 0 such that Lτ > lτ + > B + . (3.48) In view of (3.46), there exists a sequence nµ , µ = 1,2,..., such that lim yκrnµ +τ = Lτ , lim yr(κnµ −ri )+τ = Tri ,τ ≤ Lτ , µ→∞ µ→∞ (3.49) lim zr(κnµ −s j )+τ = Ss j ,τ ≥ mτ . µ→∞ In view of (3.19), (3.35), (3.46), (3.47), and (3.49), we take k ai Tri ,τ CLτ ≤B+ = Lτ Lτ = B + mi=1 j =1 b j Ss j ,τ (3.50) mτ and obviously, we have that Tri ,τ = Lτ , i = 1,2,...,k, Ss j ,τ = mτ , (3.51) j = 1,2,...,m. In addition, using (3.19), (3.35), (3.46), (3.47), and (3.51), for κ = 2, from statements (I) and (IV) of Lemma 3.5 and arguing as above, we take for γ = 0,1,..., lim yr(2nµ −w1 −2γ)+τ = Lτ , lim zr(2nµ −w1 −s1 −2γ)+τ = mτ , µ→∞ µ→∞ (3.52) and for κ = 1 and from all the statements of Lemma 3.5, lim yr(nµ −w1 −2γ)+τ = Lτ , µ→∞ lim zr(nµ −w3 −2γ)+τ = mτ , µ→∞ lim yr(nµ −w2 −2γ)+τ = Lτ , µ→∞ lim zr(nµ −w4 −2γ)+τ = mτ , (3.53) µ→∞ w1 ,w2 ,w3 ,w4 are defined in Lemma 3.5. Let a σκ ∈ {0,1,...,(3 − κ)φ}, φ = max rk ,sm . Suppose first that κ = 2. Then in view of (3.19), there exist positive integers p, q and a continuous function Fσ2 : R × R× · · · × R → R such that yr(2nµ +2σ2 )+τ = B + Fσ2 ζnµ ,0 ,... ,ζnµ ,p ,ξnµ ,0 ,...,ξnµ ,q , (3.54) where for i = 0,1,..., p, j = 0,1,..., q, ζnµ ,i = yr(2nµ −w1 −2i)+τ , ξnµ , j = zr(2nµ −w1 −s1 −2 j)+τ . (3.55) If κ = 1, there exist positive integers v1 , v2 , v3 , v4 and a continuous function Gσ1 : R × R× · · · × R → R such that yr(nµ +σ1 )+τ = B + Gσ1 ζnµ ,0 ,...,ζnµ ,v1 , ζ¯nµ ,0 ,..., ζ¯nµ ,v2 ,ξnµ ,0 ,...,ξnµ ,v3 , ξ¯nµ ,0 ,..., ξ¯nµ ,v4 , (3.56) G. Stefanidou and G. Papaschinopoulos 349 where for i = 0,1,...,v1 , i¯ = 0,1,...,v2 , j = 0,1,...,v3 , and j¯ = 0,1,...,v4 , ζ¯nµ ,i¯ = yr(nµ −w2 −2i)+τ ¯ , ξ¯n , j¯ = zr(n −w −2 j)+τ ¯ . ζnµ ,i = yr(nµ −w1 −2i)+τ , ξnµ , j = zr(nµ −w3 −2 j)+τ , µ µ (3.57) 4 Therefore, from (3.47), (3.52), (3.53), (3.54), and (3.56), it follows that lim yr(κnµ +κσκ )+τ = B + µ→∞ CLτ = Lτ . mτ (3.58) Using the same argument to prove (3.58) and using (3.19), we can easily prove that for i = 1,2,...,k, j = 1,2,...,m, lim yr(κnµ +κσκ −ri )+τ = Lτ , lim zr(κnµ +κσκ −s j )+τ = mτ . µ→∞ µ→∞ (3.59) Therefore, if δ = (mτ − C)/(Lτ − − B), then in view of (3.19), (3.47), (3.58), and (3.59), there exists a µ0 ∈ {1,2,... } such that for j = 1,2,...,m, B mτ + δ zr(κnµ0 +2φ+κ−s j )+τ ≤ C + = mτ + δ Lτ − (3.60) and so from (3.19), (3.47), (3.48), (3.58), (3.59), and (3.60), we get C Lτ − = Lτ − > l τ . yr(κnµ0 +2φ+κ)+τ ≥ B + mτ + δ (3.61) Using (3.19), (3.47), (3.48), (3.58), (3.59), and (3.61) and working inductively, we can easily prove that yr(κnµ0 +2φ+κω)+τ ≥ Lτ − > lτ , ω = 2,3,..., (3.62) which is a contradiction since liminf n→∞ yκrn+τ = lτ . Therefore, since τ is an arbitrary number such that τ ∈ {0,1,...,κr − 1}, relations (3.45) are satisfied. Moreover, from (3.19) and (3.47), we have that lim zκnr+i = ξi , n→∞ i = 0,1,...,κr − 1. (3.63) This completes the proof of the lemma. In the next proposition, we study the periodicity of the positive solutions of (1.1). Proposition 3.7. Consider (1.1), where k,m ∈ {1,2,... }, A,ci ,d j , i ∈ {1,2,...,k}, j ∈ {1,2,...,m}, are positive fuzzy numbers, and pi , i ∈ {1,2,...,k }, q j , j ∈ {1,2,...,m}, are positive integers. If (3.10) holds and r is a common divisor of the integers pi + 1, q j + 1, i = 1,2,...,k, j = 1,2,...,m, then (1.1) has periodic solutions of prime period r. Moreover, if ri , i = 1,2,...,k, (resp., s j , j = 1,2,...,m)—ri , s j are defined in (3.36)—are even (resp., odd) integers, then (1.1) has periodic solutions of prime period 2r. 350 Fuzzy diﬀerence equations Proof. From (3.10), we have that A, ci , i = 1,2,...,k, d j , j = 1,2,...,m, are positive real numbers such that (3.12) and (3.13) hold. We consider functions Li,a ,Ri,a , i = −π, −π + 1,...,0, such that for λ = 0,1,...,φ − 1, θ = 1,2,...,r, and a ∈ (0,1], L−rφ+rλ+θ,a = L−r+θ,a , R−rφ+rλ+θ,a = R−r+θ,a , (3.64) the functions Lw,a , w = −r + 1, −r + 2,...,0, are increasing, left continuous, and for all w = −r + 1, −r + 2,...,0, we have A + < Lw,a < 2A, Rw,a = ALw,a , Lw,a − A (3.65) where is a positive number such that < A. Using (3.65) and since the functions Lw,a , w = −r + 1, −r + 2,...,0, are increasing, if a1 ,a2 ∈ (0,1], a1 ≤ a2 , we get ALw,a1 Lw,a2 − A2 Lw,a1 ≥ ALw,a1 Lw,a2 − A2 Lw,a2 (3.66) which implies that Rw,a , w = −r + 1, −r + 2,...,0, are decreasing functions. Moreover, from (3.65), we get Lw,a ≤ Rw,a , A + ≤ Lw,a ,Rw,a ≤ 2A2 , (3.67) and so from [18, Theorem 2.1], Lw,a ,Rw,a , w = −r + 1, −r + 2,...,0, determine the fuzzy numbers xw , w = −r + 1, −r + 2,...,0, such that [xw ]a = [Lw,a ,Rw,a ], w = −r + 1, −r + 2,...,0. Let xn be a positive solution of (1.1) which satisfies (2.14) and let the initial values be positive fuzzy numbers such that (3.4) hold and the functions Li,a ,Ri,a , i = −π, −π + 1,...,0, a ∈ (0,1], are defined in (3.64), (3.65); Li,a , i = −π, −π + 1,...,0, a ∈ (0,1], are increasing and left continuous. Then from [16, Proposition 2], we have that for any a ∈ (0,1], the system given by (3.7), (3.12), and (3.13) has periodic solutions of prime period r, which means that there exists solution Ln,a ,Rn,a , a ∈ (0,1], of the system such that Ln+r,a = Ln,a , Rn+r,a = Rn,a , a ∈ (0,1]. (3.68) Therefore, from (2.22) and (3.68), we have that (1.1) has periodic solutions of prime period r. Now, suppose that ri , i = 1,2,...,k, (resp., si , j = 1,2,...,m) are even (resp., odd) integers. We consider the functions Li,a ,Ri,a , i = −π, −π + 1,...,0, such that analogous relations (3.37), (3.38), and (3.43) hold, Lw,a , w = −r + 1,...,0, are increasing, left continuous functions, and the first relation of (3.65) holds. Arguing as above, the solution xn of (1.1) with initial values xi , i = −π, −π + 1,...,0, satisfying (3.4), where Li,a ,Ri,a , i = −π, −π + 1,...,0, are defined above, is a periodic solution of prime period 2r. In the following proposition, we study the convergence of the positive solutions of (1.1). G. Stefanidou and G. Papaschinopoulos 351 Proposition 3.8. Consider (1.1), where k,m ∈ {1,2,... }, A,ci ,d j , i ∈ {1,2,...,k}, j ∈ {1,2,...,m}, are positive fuzzy numbers, and pi , i ∈ {1,2,...,k }, q j , j ∈ {1,2,...,m}, are positive integers. Then the following statements are true. (i) If (3.11), holds, then (1.1) has a unique positive equilibrium x and every positive solution of (1.1) nearly converges to the unique positive equilibrium x with respect to D as n → ∞ and converges to x with respect to D1 as n → ∞. (ii) If (3.10) is satisfied and r is the greatest common divisor of the integers pi + 1, q j + 1, i = 1,2,...,k, j = 1,2,...,m, such that (3.36) holds, then every positive solution of (1.1) nearly converges to a period κr solution of (1.1) with respect to D as n → ∞ and converges to a period κr solution of (1.1) with respect to D1 as n → ∞; κ is defined in (3.44). Proof. (i) Let xn be a positive solution of (1.1) which satisfies (2.14). Since (3.7) and (3.11) hold, we can apply [16, Proposition 4] and we have that for any a ∈ (0,1], there exist the limn→∞ Ln,a , limn→∞ Rn,a , and lim Ln,a = La , n→∞ lim Rn,a = Ra , n→∞ a ∈ (0,1], (3.69) where La = Al,a Ar,a − Ca Da , Ar,a − Ca k ci,l,a , d j =1 j,r,a Ca = mi=1 Ra = k Al,a Ar,a − Ca Da , Al,a − Da ci,r,a . d j =1 j,l,a (3.70) Da = mi=1 In addition, from (3.3) and (3.70), we get La ≥ B2 − Z 2 = λ, C−W Ra ≤ C2 − W 2 = µ, B−Z (3.71) where B,C (resp., Z,W) are defined in (3.3) (resp., (3.5)). Then from (3.69), (3.71), and arguing as in [13, 14, 15], we can easily prove that La ,Ra determine a fuzzy number x such that [x]a = [La ,Ra ]. Finally, using (3.70), we take that x is the unique positive equilibrium of (1.1). Using relations (3.11), (3.69), and arguing as in [15, Proposition 2], we can prove that every positive solution of (1.1) nearly converges to the unique positive equilibrium x with respect to D as n → ∞ and converges to x with respect to D1 as n → ∞. (ii) Suppose that (3.10) holds. Let xn be a positive solution of (1.1) such that (2.14) holds. Since (Ln,a ,Rn,a ) is a positive solution of the system which is defined by (3.7), (3.12), and (3.13), from Lemma 3.6, we have that lim Lκnr+l,a = l,a , n→∞ lim Rκnr+l,a = ξl,a , n→∞ a ∈ (0,1], l = 0,1,...,κr − 1, (3.72) where κ is defined in (3.44). Using (3.72) and arguing as in [15, Proposition 2], we can prove that every positive solution of (1.1) nearly converges to a period κr solution of (1.1) with respect to D as n → ∞ and converges to a period κr solution of (1.1) with respect to D1 as n → ∞. Thus, the proof of the proposition is completed. From Propositions 3.2–3.8, it is obvious that (1.1) exhibits the trichotomy character described concentratively by the following proposition. 352 Fuzzy diﬀerence equations Proposition 3.9. Consider the fuzzy diﬀerence equation (1.1), where k,m ∈ {1,2,... }, and A,ci ,d j , i ∈ {1,2,...,k}, j ∈ {1,2,...,m}, are positive fuzzy numbers. Then (1.1) possesses the following trichotomy. (i) If relation (3.1) is satisfied, then (1.1) has unbounded solutions. (ii) If (3.10) holds and r is the greatest common divisor of the integers pi + 1, q j + 1, i = 1,2,...,k, j = 1,2,...,m, such that (3.36) holds, then every positive solution of (1.1) nearly converges to a period κr solution of (1.1) with respect to D as n → ∞ and converges to a period κr solution of (1.1) with respect to D1 as n → ∞. (iii) If (3.11) holds, then every positive solution of (1.1) nearly converges to the unique positive equilibrium x with respect to D as n → ∞ and converges to x with respect to D1 as n → ∞. In the next proposition, we study the asymptotic stability of the unique positive equilibrium of (1.1). Proposition 3.10. Consider the fuzzy diﬀerence equation (1.1), where k,m ∈ {1,2,... }, A,ci ,d j , i ∈ {1,2,...,k}, j ∈ {1,2,...,m}, are positive fuzzy numbers, and pi , i ∈ {1,2,...,k}, q j , j ∈ {1,2,...,m}, are positive integers such that (3.11) holds. Suppose that there exists a positive number θ such that θ < B, Z< 2B + C − θ − (C − θ)2 + 4BC , 2 (3.73) where B,C are defined in (3.3) and Z is defined in (3.5). Then the unique positive equilibrium x of (1.1) is nearly asymptotically stable. Proof. Since (3.11) holds, from Proposition 3.8, equation (1.1) has a unique positive equilibrium x which satisfies (2.15). Let be a positive real number. Since (3.18) holds, we can define the positive real number δ as follows: δ < min{,λ,θ,B − Z }. (3.74) Let xn be a positive solution of (1.1) such that D(x−i ,x) ≤ δ ≤ , i = 0,1,...,π. (3.75) From (3.75), we have L−i,a − La ≤ δ, R−i,a − Ra ≤ δ, i = 0,1,...,π, a ∈ (0,1]. (3.76) In addition, from (3.3), (3.7), (3.74), and (3.76) and since (La ,Ra ) satisfies (3.7), we get k k ci,l,a L− pi ,a ci,l,a (La + δ) − La ≤ Al,a + mi=1 − La L1,a − La = Al,a + mi=1 j =1 d j,r,a R−q j ,a j =1 d j,r,a (Ra − δ) Ca − Al,a + La Ra − (B − Z) =δ ≤δ . Ra − δ Ra − δ (3.77) G. Stefanidou and G. Papaschinopoulos 353 From (3.74) and (3.77), it is obvious that L1,a − La < δ < . (3.78) Moreover, arguing as above, we can easily prove that R1,a − Ra ≤ δ Da − Ar,a + Ra . La − δ (3.79) We claim that θ < La − Ra + Ar,a − Da , a ∈ (0,1]. (3.80) We fix an a ∈ (0,1] and we concider the function g(h) = Al,a Ar,a − Da h Al,a Ar,a − Da h − + Ar,a − Da , Ar,a − h Al,a − Da (3.81) where h is a nonnegative real variable. Moreover, we consider the function f (x, y,z) = x2 − (2x + y)z + z2 − θ, x−z (3.82) where B ≤ x ≤ y ≤ C and W ≤ z ≤ Z, B,C (resp., W,Z) are defined in (3.3) (resp., (3.5)). Using (3.82), we can easily prove that the function f is increasing (resp., decreasing) (resp., decreasing) with respect to x (resp., y) (resp., z) for all y,z (resp., x,z) (resp., x, y) and so from (3.73), f (x, y,z) > f (B,C,Z) = B 2 − (2B + C)Z + Z 2 − θ > 0. B−Z (3.83) Therefore, from (3.3), (3.81), (3.82), and (3.83), we have g(0) = f Al,a ,Ar,a ,Da + θ > θ. (3.84) In addition, from (3.81), we can prove that g is an increasing function with respect to h and so we have g(0) < g(Ca ), a ∈ (0,1]. Therefore, from (3.70), (3.81), and (3.84), relation (3.80) is true. Hence, from (3.74), (3.79), and (3.80), we get R1,a − Ra < δ < . (3.85) From (3.7), (3.76), (3.78), and (3.85) and working inductively, we can easily prove that Ln,a − La ≤ , Rn,a − Ra ≤ , a ∈ (0,1], n = 0,1,..., (3.86) 354 Fuzzy diﬀerence equations and so n ≥ 0. D xn ,x ≤ , (3.87) Therefore, the positive equilibrium x is stable. Moreover, from Proposition 3.8, we have that every positive solution of (1.1) nearly tends to x with respect to D as n → ∞. So, x is nearly asymptotically stable. So, the proof of the proposition is completed. Finally, we study the oscillatory behavior of the positive solutions of the fuzzy diﬀerence equation k c2s+1 xn−2s−1 , s=0 d2s+2 xn−2s xn+1 = A + s=k 0 (3.88) where k is a positive integer, and A, c2s+1 , d2s+2 , s ∈ {0,1,...,k}, are positive fuzzy numbers. Obviously, (3.88) is a special case of (1.1). In what follows, we need to study the oscillatory behavior of the positive solutions of the system of ordinary diﬀerence equations k a2s+1 yn−2s−1 , s=0 b2s+2 zn−2s yn+1 = B + s=k0 k zn+1 = C + s=0 e2s+1 zn−2s−1 , k s=0 h2s+2 yn−2s n = 0,1,..., (3.89) where k is a positive integer, B,C,a2s+1 ,b2s+2 ,e2s+1 ,h2s+2 , s ∈ {0,1,...,k}, are positive real constants, and the initial values y j , z j , j = −2k − 1, −2k,...,0, are positive real numbers. Let (yn , zn ) be a positive solution of (3.89). We say that the solution (yn , zn ) oscillates about (y, z), y,z ∈ R+ , if for every n0 ∈ N, there exist s, m ∈ N, s, m ≥ n0 , such that ys − y ym − y ≤ 0, ys − y zs − z ≥ 0, zs − z zm − z ≤ 0, ym − y zm − z ≥ 0. (3.90) Lemma 3.11. Consider system (3.89), where k is a positive integer, B, C, a2s+1 , b2s+2 , e2s+1 , h2s+2 , s ∈ {0,1,...,k}, are positive real constants, and the initial values y j , z j , j = −2k − 1, −2k,...,0, are positive real numbers.A positive solution yn ,zn of system (3.89) oscillates about the unique positive equilibrium x̄, ȳ of system (3.89) if either the relations Λ ≥ max Λ1,s ,Λ2,s , ∆ ≥ max ∆1,s ,∆2,s , s = 0,1,...,k, (3.91) s = 0,1,...,k, (3.92) or the relations Λ ≤ min{Λ1,s ,Λ2,s , ∆ ≤ min{∆1,s ,∆2,s , G. Stefanidou and G. Papaschinopoulos 355 hold, where for s = 0,1,...,k, Λ= ∆1,s = h2s+2 e2s+1 Λ2,s = a2s+1 y−2s−1 , s=0 b2s+2 z−2s ∆ = s=k0 1 1 s=0 e2s+1 z−2s−1 , k s=0 h2s+2 y−2s s k s −1 k ȳ µ b2 j+2 z̄+ b2 j+2 z−2 j+2+2s − a2 j+1 ȳ+ a2 j+1 y−2 j+1+2s z̄ j =0 j =s+1 j =0 j =s+1 a2s+1 ∆2,s = Λ1,s = 1 k k 1 b2s+2 −1 k s −1 k ȳ s e2 j+1 z̄ + e2 j+1 z−2 j+2s − h2 j+2 ȳ + h2 j+2 y−2 j+1+2s λz̄ j =0 j =s j =0 j =s+1 s −1 k s −1 k z̄ a2 j+1 ȳ+ a2 j+1 y−2 j+2s − b2 j+2 z̄ + b2 j+2 z−2 j+1+2s µ ȳ j =0 j =s j =0 j =s+1 k − B, − B, s k s −1 k z̄ λ h2 j+2 ȳ+ h2 j+2 y−2 j+2+2s − e2 j+1 z̄+ e2 j+1 z−2 j+1+2s ȳ j =0 j =s+1 j =0 j =s+1 − C, − C, k e2s+1 , s=0 h2s+2 a2s+1 . s=0 b2s+2 λ = ks=0 µ = sk=0 (3.93) Proof. Suppose that (3.91) hold. We prove that for ρ = 0,1,...,k, y2ρ+1 ≥ ȳ, z2ρ+1 ≥ z̄, y2ρ+2 ≤ ȳ, z2ρ+2 ≤ z̄. (3.94) From (3.89) and (3.91), we have k a2s+1 y−2s−1 = B + ∆ ≥ B + ∆1,k = ȳ, s=0 b2s+2 z−2s y1 = B + s=k0 (3.95) z1 = C + Λ ≥ C + Λ1,k = z̄. Since from (3.91), Λ ≥ Λ2,0 and ∆ ≥ ∆2,0 , then from (3.89), we have k y2 = B + s=0 a2s+1 y−2s b2 z1 + ks=1 b2s+2 z1−2s µ ȳ (C + Λ)b2 + ks=1 b2s+2 z1−2s µ ȳ ≤B+ =B+ = ȳ, k z̄ z̄ b2 z1 + s=1 b2s+2 z1−2s z2 ≤ C + λz̄ = z̄. ȳ (3.96) Using (3.89), (3.91), (3.95), and (3.96), relations ∆ ≥ ∆1,ρ−1 , Λ ≥ Λ1,ρ−1 (resp., ∆ ≥ ∆2,ρ , Λ ≥ Λ2,ρ ), ρ = 1,2,...,k, and working inductively, we can easily prove (3.94) for ρ = 1,2,...,k: y2ρ+1 ≥ ȳ, z2ρ+1 ≥ z̄ resp., y2ρ+2 ≤ ȳ, z2ρ+2 ≤ z̄ . (3.97) 356 Fuzzy diﬀerence equations Therefore, (3.94) hold for ρ = 0,1,...,k. Then since (3.94) hold for ρ = 0,1,...,k, using (3.89) and working inductively, we can easily prove that(3.94) hold for any ρ = k + 1,k + 2,..., and so if (3.91) hold, the proof of the lemma is completed. Similarly, if (3.92) are satisfied, then we can easily prove that y2ρ+1 ≤ ȳ, z2ρ+1 ≤ z̄, y2ρ+2 ≥ ȳ, z2ρ+2 ≥ z̄, ρ = 0,1,.... (3.98) This completes the proof of the lemma. Using Lemma 3.11 and arguing as in [13, Proposition 2.4], we can easily prove the following proposition which concerns the oscillatory behavior of the positive solutions of the fuzzy diﬀerence equation (3.88). Proposition 3.12. Consider (3.88), where k is a positive integer, and A, c2s+1 , d2s+2 , s ∈ {0, 1,...,k}, are positive fuzzy numbers. Then a positive solution xn of (3.88) satisfying (2.14) oscillates about the positive equilibrium x, which satisfies (2.15) if, for any s = 0,1,...,k and a ∈ (0,1], either the relations Λ̄a ≥ max Λ̄1,s,a , Λ̄2,s,a , ∆¯ a ≥ max ∆¯ 1,s,a , ∆¯ 2,s,a (3.99) or the relations Λ̄a ≤ min Λ̄1,s,a , Λ̄2,s,a , ∆¯ a ≤ min ∆¯ 1,s,a , ∆¯ 2,s,a (3.100) hold, where Λ̄a , ∆¯ a , Λ̄1,s,a , Λ̄2,s,a , ∆¯ 1,s,a , ∆¯ 2,s,a are defined for the analogous system (3.7) in the same way as Λ, ∆, Λ1,s ,Λ2,s , ∆1,s ,∆2,s were defined in Lemma 3.11 for system (3.89). Using Proposition 3.12, we take the following corollary. Corollary 3.13. Consider (3.88), where k is a positive integer, and A, c2s+1 , d2s+2 , s ∈ {0,1,...,k }, are positive fuzzy numbers. Then a positive solution xn of (3.88) satisfying (2.14) oscillates about the positive equilibrium x, which satisfies (2.15) if, for any p = 0,1,...,k and a ∈ (0,1], either the relations L−2k−1+2p,a ≥ La , L−2k+2p,a ≤ La , R−2k−1+2p,a ≥ Ra , R−2k+2p,a ≤ Ra (3.101) or the relations L−2k−1+2p,a ≤ La , L−2k+2p,a ≥ La , R−2k−1+2p,a ≤ Ra , R−2k+2p,a ≥ Ra hold. Acknowledgment This work is a part of the first author Doctoral thesis. (3.102) G. Stefanidou and G. Papaschinopoulos 357 References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] R. P. Agarwal, Diﬀerence Equations and Inequalities, Monographs and Textbooks in Pure and Applied Mathematics, vol. 155, Marcel Dekker, New York, 1992. D. Benest and C. 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